Some new results on interval-valued volterra integro-differential equations for caputo fractional derivative

Abstract

Volterra integro differential equations are one of the necessary tools to discuss the framework of modeling systems with memory and nonlinear behaviors comprehensively, which commonly appear in several engineering and scientific applications. For accurate results, researchers incorporate fractional calculus approach in mathematical models models as compared to classical order derivatives and integrals. This research article presents some novel results of Caputo fractional-order linear Volterra integro-differential equations based upon the concepts of gH derivative for the latest concept of interval-valued functions. The main objective of this research is to obtain the unique solutions of interval-valued Volterra integro-differential equations. We have provided some proofs and examples to illustrate the novelty of our obtained results. Also, graphical representations demonstrate the behavior of solutions under different initial conditions for researcher’s understanding.

Introduction

Integro-differential equations play a vital part in mathematics to study real-life problems, especially in constrained model¹. This article is motivated to study the integro-differential equation on a novel concept of interval-valued functions (IVF) under fractional order calculus. The study of Interval-valued Volterra integro-differential equations (VIDE) in the Caputo sense has received a lot of attention because of their ability to model a wide range of complex engineering, biological, and physical phenomena. The innovation of utilizing the Caputo fractional derivative (CFD) resides in its capacity to generalize classical differential equations while maintaining initial conditions congruent with conventional integer-order derivatives. The vast majority of real-world issues in disciplines like material science, engineering mechanics, and thermal analysis involve some uncertainty. Environmental effects, inaccurate measurements, or lack of knowledge can all contribute to this uncertainty. It might be required in some situations to model some variables as fuzzy numbers and others as intervals.

The classical concepts of integer-ordered integrals and derivatives are extended to noninteger-ordered integrals in fractional calculus (FC). FC has applications in various fields such as physics^2,3, wave theory⁴, signal processing^5,6, epidemiology^7,8, and control systems^9,10. For an extensive overview of FC, the reader can study the books of Samko and Kilbas¹¹, Podlubny¹², and Diethelm¹³. Recent studies expanded the applications of FC with more recent work like¹⁴ studying the solution of piecewise equations with the Caputo-Fabrizio derivative and proving the solution’s existence, uniqueness, and Hyers–Ulam stability using Banach’s and Krasnoselskii’s fixed-point theorems. In Ref.¹⁵, the authors used the Atangana–Baleanu–Caputo derivative with a Mittag–Leffler kernel to solve delay impulsive implicit fractional differential equations and showed the resolution of the problem using Gronwall’s and Banach–Schaefer fixed-point theorems. In¹⁶, the researchers analysed the transmission dynamics of Nipah virus (NiV) using piecewise fractional operators and implemented analytical and numerical methods to represent multi-step crossover behavior. The authors obtained solutions of coupled delay fractional systems with nonlocal boundary conditions by demonstrating existence, uniqueness, and Ulam–Hyers stability with provided illustrative examples in Ref.¹⁷. The authors studied a coupled system with generalized proportional Riemann–Liouville derivatives and integral boundary conditions, applying Schauder’s and Banach’s theorems with numerical validation in Ref.¹⁸. In Ref.¹⁹, the authors studied the Caputo fractional problem with a p-Laplacian operator, from which they derived Green’s function and proved the corresponding stability results by means of topological degree theory. Hilfer hybrid equations with delay terms and applying multiple fixed-point theorems with supporting examples are studied in Ref.²⁰. The authors in Ref.²¹ studied implicit ABC fractional equations of orders 2-3 and established existence and stability properties. The qualitative conditions for a pantograph system with ABC derivatives and integral constraints were formulated in Ref.²². Initially, Agarwal, Lakshmikansham, and Nieto introduced the idea of using ambiguous parameters for fractional differential equations²³. Lupulescu and O’Regan have presented a novel idea of differentiability for IVF²⁴. In addition, Lupulescu has discussed a comprehensive study of FC for IVF^25,26. Various researchers introduced definitions for fractional order derivatives like Riemann–Liouville (RLFD), Caputo (CFD), Riesz (RDF), Grunwald–Latnikov (GLFD), etc. When dealing with real-world problems, CFD is the appropriate choice. The main benefit of using the Caputo approach is that it allows the initial and boundary conditions for differential equations, which can be interpreted similarly for integer-order differential equations.

The importance of the interval analysis is well known theoretically and in terms of its applications. Interval analysis is an effective mathematical tool in various fields such as mathematics, signal processing, computer science, and forecasting. Interval analysis was introduced to deal with ambiguity in many computer and mathematical models. Moore introduced the concept of interval analysis²⁷. For example,

1

is an IVF and is graphically represented as

In Fig. 1, the shaded region shows the range of lower and upper bounds of , that is, and , respectively, as mentioned in Eq. (1).

Fig. 1.

Graph of IVF .

Solutions through tangency and stacking theorems for set-valued differential equations can be seen in Fréchet spaces with semimetrics and Hukuhara derivatives²⁸, along with framed in Ref.²⁹ and other works, and for Ref.³⁰ that addressed uncertain nonlinear interval problems using deterministic transformations and genetic algorithms. In 1967, Hukuhara introduced a novel concept of difference for IVF known as the “Hukuhara difference” and the “Hukuhara derivative”³¹. Markov presented various results for different arithmetic operations along with some basic properties³². Stefanini and Bede³³ suggested a new definition of the generalized Hukuhara difference for the intervals denoted by the gH-difference, which presents some properties of the proposed concepts.

Malinowski presented the concept of the Hukuhara derivative in his research on interval-valued differential equations³⁴. A famous researcher, Chalco-Cano^35,36 studied linear interval-valued differential equations and derived their solutions with a gH-derivative approach. Tao discussed some of the most important characteristics of interval-valued functions (IVF), particularly continuity, integration properties, and differentiability³⁷. In Ref.³⁸, the author used the fractional differential transform method (FDTM) to solve fuzzy fractional differential equations. The resulting generalized solutions have broad applications in science and engineering. In Ref.³⁹, HATM is used to find fuzzy analytical solutions for fractional differential equations. The method reliably produces symmetric upper and lower bound solutions for applications in mathematics and engineering.

One major class of integro-differential equations is the Volterra type. VIDE is named after Volterra. There are two types of VIDE, kind I and kind II. A wide range of physical applications regarding linear and non-linear VIDE arises, including nano hydrodynamics, heat transfer, diffusion processes, biological species, control theory, neural networks, etc., and they provide a suitable mathematical approach for solving complex real-life problems. VIDE gives accurate results according to their homogeneous or non-homogeneous types⁴⁰. In the last decade, the solutions of two-dimensional VIDE have been a topic for researchers^41,42. A comprehensive overview of solutions of different forms of interval-valued VIDE for classical order was presented by A. Younus⁴³. In our investigation, we are motivated to study fractional order VIDE for IVF, as FC is a powerful tool for interval differential equations. This research aims to develop some results for fractional order VIDE with the Caputo approach. From this point of view, this research will open a new direction to enhance the applications of fractional order VIDE.

This article is organized as follows: “Preliminaries” section contains some basic definitions and preliminaries. Main results of interval-valued VIDE under CFD (FVIDE) have been discussed in “Interval-valued Volterra integro-differential equations for caputo fractional derivative” section. Some examples have also been presented for illustration.

Also, throughout this article, we will use for the Caputo derivative and for fractional integral (for our convenience).

Novelty of our work

In this article, we present some new results for VIDE defined on interval-valued functions, which appear in several real-life phenomena. Our results are generalized due to the fractional order calculus, which is more effective for accurate results of our problem as compared to the classical approach. A fractional derivative provides a robust framework for modeling systems with uncertain behaviors as compared to the classical one. We have obtained the solutions in two forms and discussed them through examples and graphs.

Preliminaries

A comprehensive overview of some basic concepts, well-known results, and notations is presented in this section. These concepts will help to understand the further discussion. Here, represents the set of all nonempty bounded and closed intervals of the real line, and AC[a, b] denotes the space of absolutely continuous functions on the interval [a, b]. Thus, represents the space of all interval-valued absolutely continuous functions.

Definition 2.1

²⁵ Let is an IVF and , and . Then the interval-valued Riemann-Liouville fractional integral of X is defined as

2

In addition, it is clear that can be written as

Definition 2.2

²⁵ If and , then

3

and is called interval-valued CFD.

Definition 2.3

⁴⁴ The CFD of a constant function is zero. If an IVF is a constant function then

Remark 2.1

²⁵ Let , and , then

If X is -monotone, then

• (i)If X is -increasing, then it follows .
• (ii)If X is -decreasing, then it follows .

Theorem 2.1

²⁵ If is a -monotone IVF and , then

4

Theorem 2.2

²⁵Let be such that either X is -increasing (or-decreasing) on [a, b] and is -increasing on [a, b], then

5

Theorem 2.3

²⁵ Let be -monotone on [a, b], then for a.e.

• (i)
  Let X and Y are equally

  6

  and

  7

  Moreover, if the difference has constant sign, then

  8
• (ii)
  If X and Y are differently [a, b], then

  9

  and

  10

  Moreover, if the difference has constant sign, then

  11

Definition 2.4

^32,33 Let the set of all non-empty closed and bounded intervals of the real line be denoted by . Let , , then the Minkowski addition and scalar multiplication are given as

and

In Minkowski addition the opposite of E is not the inverse of E (unless is a singleton). The Minkowski difference for the intervals and is defined as . Hence, is a quasilinear space according to the above operation³².

The generalized Hukuhara difference for the intervals and is defined as

12

Example 2.1

Let and ,

by equation (12), we can write

Definition 2.5

The width of E denoted by is defined as³⁶

, where and

13

In general, .

Suppose is an IVF, then exists if and only if and exist. So,

and F(s) is continuous iff and are continuous.

Definition 2.6

(Generalized Hukuhara Derivative)^25,33 Let is an IVF and let . The gH-derivative of F(s) is given by

Definition 2.7

²⁵ An IVF is -increasing (or -decreasing) on [a, b] if the real function is increasing or decreasing. If F is -increasing (or -decreasing) then F is called -monotone.

Theorem 2.4

³⁵ If then iff

(a) and The converse part of theorem

(2.4)(a) is not true, that is, the g-differentiability of IVF FH differentiability of and .

(b) exists and satisfy

and

Furthermore,

Interval-valued Volterra integro-differential equations for caputo fractional derivative

We demonstrate some results about the uniqueness of solutions in analyzing the behavior of interval-valued VIDE with the Caputo approach. We will use the basic definitions needed for our analysis of the main result.

Theorem 3.1

The linear nonhomogeneous interval-valued FVIDE is defined as

14

where initial condition is The assumptions are , , , is an IVF such that , is an IVF and gH-differentiable function, and are real-valued functions, and are also real-valued and continuous functions, then the unique solution of Eq. (14) is written as

15

satisfying the initial condition and .

Proof

Applying the fractional integral on the Eq. (14), we have

16

Using Theorem 2.1, the Eq. (16) can be written as

or

or

17

solving

18

19

and

20

Substituting Eqs. (18), (19), and (20) in Eq. (17), we obtained the solution of Eq. (14) given in Eq. (15).

Remark 3.1

Let , where . The length of is defined by³⁶

If is a constant function then by⁴⁴,

Example 3.1

then

21

and at

22

applying on Eq. (22), which implies that

Since the CFD of a constant function is zero by Remark 3.1,

then from Eq. (21) one can easily calculate

implies that

and implies that

Proposition 3.1

If is a solution of the nonhomogeneous linear Interval-valued FVIDE equation, defined as

23

with , and then P(.) satisfies Eq. (15).

Proof

From Eq. (15), we have

24

and

We can write

then from Theorem 2.2 and Remark 3.1, we can write

So, is the solution of the interval-valued FVIDE (23).

Corollary 3.1

If P(.) satisfies Eq. (15), then P(.) with is a solution of following interval-valued FVIDE

25

Theorem 3.2

Consider the linear non-homogeneous interval-valued FVIDE given by

26

under all the assumptions as using for Eq. (15) and is -increasing. Then the solution of interval-valued FVIDE (15) can be classified into the following parts:

• (i)
  

  for and
• (ii)
  

  for and
• (iii)
  

  for and
• (iv)
  

  for and

Proof

• (i)
  Let and , applying fractional integral on Eq. (26), then

  27

then from Theorem 2.1, we can write Eq. (27) as

or

or

or

28

Similar to part (i), solutions of parts (ii), (iii), and (iv) can be found by considering the same assumptions.

Example 3.2

Consider the interval-valued FVIDE (14), particularly for , , , , then

29

and for equation (29) becomes

30

Since both and are increasing, so from Remark (2.1, Part(i)), Eq. (30) can be written as

31

and

32

solving both Eqs. (31) and (32) with the solution of Eq. (30) is

33

For the reader’s convenience, Table 1 contains the solutions of the different values of for the Eq. (29) under the given assumptions.

Table 1.

Solutions for various values of in Eq. (29).

The Figs. 2, 3, 4, 5, 6, 7, 8 and 9 illustrate the solutions of the equations as mentioned in Table 1 for respectively, and the reader can observe the increasing trend in the upper and lower bounds, i.e., increases with , and one can easily understand the effect of the fractional parameter .

Fig. 2.

Equation (33) for

Fig. 3.

.

Fig. 4.

.

Fig. 5.

.

Fig. 6.

.

Fig. 7.

.

Fig. 8.

.

Fig. 9.

.

In Figs. 10, 11, 12, 13 and 14 these illustrate the behaviors of IVF in the different domain of particularly for only in Eq. (33).

Fig. 10.

.

Fig. 11.

.

Fig. 12.

.

Fig. 13.

.

Fig. 14.

.

Example 3.3

Consider the interval-valued FVIDE (14), particularly for , , , and , then

34

and for Eq. (34) takes the form

35

Since both and are increasing, so from Remark (2.1, Pars(i)), Eq. (35) can be written as

36

and

37

Solving both Eqs. (36) and (37) with the solution of Eq. (35) is

38

For the reader’s convenience, Table 2 contains the solutions of the different values of for the Eq. (34).

Table 2.

Solutions for various values of .

The Figs. 15, 16, 17, 18, 19, 20, 21 and 22 illustrate the solutions of the equations as mentioned in Table 2 respectively, and the reader can observe the increasing trend in the upper and lower bounds, i.e., increases with . Also, from the Figs. 15, 16, 17, 18, 19, 20, 21 and 22 and in view of Table 2, one can easily understand the effect of the fractional parameter .

Fig. 15.

Solution of Eq. (38) for .

Fig. 16.

.

Fig. 17.

.

Fig. 18.

.

Fig. 19.

.

Fig. 20.

.

Fig. 21.

.

Fig. 22.

.

In Figs. 23, 24, 25, 26 and 27, these illustrate the behaviors of IVF in the different domains of , particularly for , only in Eq. (38).

Fig. 23.

.

Fig. 24.

.

Fig. 25.

.

Fig. 26.

.

Fig. 27.

.

Example 3.4

The interval-valued FVIDE (14) can be written particularly for , , , and as

39

after solving Eq. (39) for , the solution is of the form

40

For the reader’s convenience, Table 3 contains the solutions of the different values of .

Table 3.

Solutions for various values of .

	)

The Figs. 28, 29, 30, 31, 32, 33, 34 and 35 illustrate the solutions of the equations mentioned in Table 3 respectively, and the reader can observe the increasing trend in the upper and lower bounds, i.e., increases with . Also, from the Figs. 28, 29, 30, 31, 32, 33, 34 and 35 and because of Table 3, one can easily understand the effect of the fractional parameter .

Fig. 28.

Equation (40) for .

Fig. 29.

.

Fig. 30.

.

Fig. 31.

.

Fig. 32.

.

Fig. 33.

.

Fig. 34.

.

Fig. 35.

.

In Figs. 36, 37, 38, 39 and 40 these illustrate the behaviors of IVF in the different domains of , particularly for , only in Eq. (40).

Fig. 36.

.

Fig. 37.

.

Fig. 38.

.

Fig. 39.

.

Fig. 40.

.

Conclusion

In this article, we have obtained the solutions of linear interval-valued VIDE of several kinds within the context of Caputo’s Fractional Derivative. Demonstrating the potential of interval-valued functions, this study offers a strong framework for handling various types of ambiguities in computational models which appear in control theory, artificial intelligence, numerical analysis and many other real-life situations. In this work, the derived results highlight more precision and flexibility of IVFs for due to the involved fractional approach. It is also important to note that our novel results are the generalization of classical order results obtained in⁴³. In future, this research can provide a framework for nonlinear interval-valued VIDE to explore high-dimensional problems in computational optimization and this research will be helpful to address the applications which can be solved under numerical techniques and machine learning methodologies to facilitate more researchers. Further, the interval-valued VIDE can be studied for newly defined fractional calculus⁴⁵ in the framework of AB and Caputo-Fabrizio operators and weighted fractional operators⁴⁶ with respect to another function via modified generalized Mittag-Leffler law.

Author contributions

Saima Noureen has done the research and all results are novel made by her. Anum Zehra is the research supervisor of Saima Noureen, so they both are equal contributor in this research task. Tosala has guided about the analytical aspects of fractional calculus which are main part of this latest research work.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.